Summation equations

VVe therefore return to the point-charge model for calculating electrostatic interactions. If sufficient point charges are used then all of the electric moments can be reproduced and the multipole interaction energy. Equation (4.30), is exactly equal to that calculated from the Coulomb summation. Equation (4.19). 

Sunune, /. sum, amount, summen, v.t. hum. buzz. — v.t. sum up, add. Summen-gleichung, /. summation equation, -regel, /. sum rule, rule of sums, -satz, m. principle or law of sums, -wirkung, /, combined action or effect. 

In addition to the equations arising from the material and energy balances over a stage, and the equilibrium relationships, there will be a fourth relationship, the summation equation for the liquid and vapour compositions ... 

Using a syllogistic approach analogous to the earlier construction of the Pshg perturbation summation (Equation 4), we can "derive" the general third order perturbation theory expression(l),... 

Cpm - is the standard heat capacity of species j at constant pressure, and Ns is the number of species in the chemical reaction. Note that the standard molar heat capacities of the elements cancel in the summation. Equation 10.4-2 was given earlier as equation 3.2-19, and it was used to show that the pK of acetic acid goes through a maximum not far from room temperature. Equation 10.4-3 was given earlier as equation 3.2-16. 

Half-reactions can be added to produce a net reaction, which is the oxidation-reduction reaction. However, this summation cannot be performed unless the electron numbers are the same on both sides of the reaction by agreement among chemists, electrons are not written into summation reactions. The way in which adjustments are made is to preserve the ratio of coefficients in the individual balanced half-reaction by multiplying all of the participants in an equation by the same number. The goal is to have the same number of electrons on opposite sides of the half-reactions. The electrons will then algebraically cancel when the half-reactions are added. Since the summation equation should not have coefficients divisible by a common factor, it is customary to choose numbers that will yield the least number of electrons for cancellation. 

From one point of view, the Cy are excellent choices as host molecules because they absorb only in the far UV. As a result, the induced CD signal in the near UV is due only to the complex and its magnitude is, in principle, directly proportional to the concentration of the complex in solution, ipi = 0. d [Cy-L]. This is really true only in the cases where L is an inherently achiral molecule. For racemic mixtures the more exact Beers law summation equation that relates the observed induced signal to the individual contributions is ... 

The summation equation. The Summation equation or composition constraint simply states that the sum of the mole fractions on each stage is equal to unity. For the liquid phase,... 

Errors in the MESH equations of Sec. 4.1.2 should be small, including the stage energy total material and component balances and summation equation should be small. The physical solution criteria above should take precedence over any mathematical criteria, such as having Newton-Raphson functions approach zero (Sec. 4.2.6). 

The BP methods use a form of the equilibrium equation and summation equation to calculate the stage temperatures, The first BP method, by Wang and Henke (24), included the first presentation of the tridiagonal method to calculate the component flow rates or compositions. These are used to calculate the temperatures by solving the bubble-point equation but this temperature calculation can be prone to failure. 

The Tomich method. This method has the summation equations for the vapor and liquid compositions are merged for the first independent equation ... 

Tne MESH Equations (the 2c + 3 Formulation) The equations that model equilibrium stages often are referred to as the MESH equations. The M equations are the material balance equations, E stands for equilibrium equations, S stands for mole fraction summation equations, and H refers to the heat or enthalpy balance equations. 

Since the total vapor and liquid flow rates are, by definition, the sum of the component flow rates of the respective phases, the summation equations and total mass balance equation are satisfied automatically. Thus, the number of equations and variables per stage has been reduced by 2 to 2c + 1. 

In addition, we need summation equations for the mole fractions in the vapor and liquid phases. 

Eer stage. As with the equilibrium-stage model discussed above, we ave not included the feed mole fraction summation equation, or those for the vapor and hquid streams coming from adjacent stages. 

The overall transport equation for the multicomponent continuum is then obtained by summing over components, a procedure which is in accord with the idea of independent coexistent continua. We choose at the arbitrary time t all of the control volumes to be coexistent, that is, = t for all K. We henceforth refer to a volume t thus defined as being of the multicomponent continuum at time t. After summation, equation (8) becomes... 

A trial calculation illustrates the nature of the solution. Vapor pressures are taken from Example 3a or 3fc a trial value of %i then allows calculation of Yi and Ys by Eqs. B) and (C) of Example 3. The values of Xi, K, and P that result are substituted into the summation equation. In the unlikely event that the sum is indeed unity, the chosen value of %i is correct. If not, then successive trials easily lead to this value. Note that the trivial solution giving E= 1 must be avoided. More elegant solution procedures can of course be employed. The answers are... 

---